Multi-Period Facility Location - Location Science - page 287

total demand. In the multi-period case, these constraints are written as

0
@ Q i X
2T it

1
A X
j2J

X

d jt ; t 2 T;

z i (11.53)

i2I

which can be easily accommodated in ( 11.52 ).
For the linear relaxation of model ( 11.41 )-( 11.45 ), ( 11.51 ), and ( 11.52 ), Sal-
danha da Gama ( 2002 ) extended the dual-based procedure proposed by Van Roy and
Erlenkotter ( 1982 ), thus obtaining sharp lower and upper bounds for the problem.
The inclusion of capacity constraints is an important step towards building more
comprehensive multi-period facility location models. Nevertheless, the capacity
constraints ( 11.51 ) are rather restrictive when it comes to real applications, namely
those arising in logistics (see Chap. 16 ). By considering a fixed capacity in each
location, these constraints neglect the possibility of making future adjustments in
the capacity of the facilities, which is a feature quite relevant in practice. In fact, it is
often the case that adjusting the capacity of an existing facility is more advantageous
from a cost point of view than installing a new facility in some other location. One
attempt to overcome such restrictive representation for the capacities was made
by Van Roy and Erlenkotter ( 1982 ) who considered exogenous time-dependent
capacities Q it (i 2 I, t 2 T ). Nevertheless, this is still unsatisfactory from a
practical point of view because no connection is established between the capacities
in different periods.
The problem of planning for the capacity expansion of existing facilities was
very much in focus in the 1970s and in the 1980s (see, for instance, Erlenkotter
1981 , and Lee and Luss 1987 ). However, at that time, the focus was put mainly on
the expansion of existing facilities. In many cases, the location of facilities was not
even a decision to make. Furthermore, many of these works considered continuous
adjustments in the capacities, which is often not adequate from a practical point of
view. In fact, if we think of production or sorting lines, we immediately realize that
changes in the capacities should be modular, or at least discrete.
One paper that clearly interconnects multi-period facility location decisions with
discrete capacity expansion is due to Shulman ( 1991 ). A set of facility types P
is considered, and in each location, facilities of different types can be progressively
established during the planning horizon, as a way of adjusting the operating capacity
of the system. In each period, at most one facility of each type can be installed in
each location but several facilities can be installed if they are of different types. For
each location i 2 I,asetP i P is assumed, corresponding to the set of facility
types that can be located at i. Denote by c ijpt the cost for supplying all the demand
of customer j 2 J in period t 2 T from a facility operating at i 2 I that is of
type p 2 P i .Letf ipt be the cost for installing a facility of type p 2 P i at i 2 I
in period t 2 T . Additionally, let Q p be the capacity of a facility of type p 2 P.
Finally, let n ip 0 denote the number of facilities of type p 2 P i operating at location
i 2 I before the beginning of the planning horizon (i.e., the problem captures the
situation in which the system is not built from scratch but is to be adapted to future